Abstract
We study the homeomorphic embeddings of a compact set K, a union of nondegenerate continua, into \(\overline {\mathbb{R}} ^n\) which preserve the conformal moduli of all condensers whose plates are continua in K. Using a result by V. N. Dubinin together with the estimates for the conformal moduli of infinitesimal condensers, we prove that Belinskii's conjecture (that such a mapping extends to a Mobius automorphism of the whole space \(\overline {\mathbb{R}} ^n2\)), corroborated by the author in 1990 for n=2 is also valid for n>2 if the compact set in question is regular at some collection of (n+2) points. This essentially strengthens the previous result of the author (1992) in which regularity was required at each point of the compact set.
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Aseev, V.V. Deformation of Plates of Small Condensers and Belinskii's Problem. Siberian Mathematical Journal 42, 1013–1025 (2001). https://doi.org/10.1023/A:1012836124090
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DOI: https://doi.org/10.1023/A:1012836124090